Missing data and multiple imputation
MACS 30200
University of Chicago
Causes of missingness
- Surveys
- Errors in data collection
- Intentional
- Censored values
Patterns of missingness
- Missing completely at random (MCAR)
- Missing at random (MAR)
- Missing not at random (MNAR)
- Why do we care?
- Mechanism
- Ignorable vs. non-ignorable
Things to consider
- Does the method provide consistent estimates of the population parameters?
- Does the method provide valid statistical inferences?
- Does the method use the observed data efficiently or does it recklessly discard information?
Complete-case analysis
- Listwise deletion
- Advantages
- Disadvantages
Available-case analysis
- Pairwise deletion
- Advantages
- Disadvantages
Imputation
- Imputation
- Unconditional mean imputation
- Conditional mean imputation
Maximum-likelihood estimation
\[p(\mathbf{X}, \theta) = p(\mathbf{X}_{\text{obs}}, \mathbf{X}_{\text{mis}}; \theta)\]
Data MAR
\[p(\mathbf{X}_\text{obs}; \theta) = \int{p(\mathbf{X}_{\text{obs}}, \mathbf{X}_{\text{mis}}; \theta)} d\mathbf{X}_{\text{mis}}\]
- EM algorithm
Single model for all variables
Predictive mean matching
- For cases with no missing data, estimate a linear regression of \(x\) on \(z\), producing a set of coefficients \(b\)
- Make a random draw from the posterior predictive distribution of \(b\), producing a new set of coefficients \(b*\)
- Using \(b*\), generate predicted values for \(x\) for all cases
- For each case with missing \(x\), identify a set of cases with observed \(x\) whose predicted values are close to the predicted value for the case with missing data
- From among those close cases, randomly choose one and assign its observed value to substitute for the missing value
Non-parametric models
- Random forests
- Deep learning
Multiple imputation
- Generate multiple imputed datasets
- Bayesian multiple imputation
- Account for uncertainty within and across the datasets
Conducting inference
\[\tilde{\beta}_j \equiv \frac{\sum_{l=1}^g B_j^{(l)}}{g}\]
Conducting inference
\[\tilde{\text{SE}}(\tilde{\beta}_j) = \sqrt{V_j^{(W)} + \frac{g + 1}{g} V_j^{(B)}}\]
\[V_j^{(W)} = \frac{\sum_{l=1}^g \text{SE}^2(B_j^{(l)})}{g}\]
\[V_j^{(B)} = \frac{\sum_{l=1}^g (B_j^{(l)} - \tilde{B}_j)^2}{g-1}\]
\[\text{SE}^2(B_j^{(l)})\]
Practical considerations for multiple imputation
- Which variables to include
- Transform variables to approximately normal
- Adjust the imputed data to resemble the original data
- Make sure the imputation model captures relevant features of the data
- \(g\) doesn’t need to be large
Infant mortality
MI scatterplot
Conducting inference
## # A tibble: 20 × 6
## id term estimate std.error statistic p.value
## <chr> <chr> <dbl> <dbl> <dbl> <dbl>
## 1 imp1 (Intercept) 6.47310 0.16284 39.75 1.30e-96
## 2 imp1 log(GDPperCapita) -0.20206 0.02988 -6.76 1.47e-10
## 3 imp1 contraception -0.00480 0.00241 -2.00 4.72e-02
## 4 imp1 educationFemale -0.14254 0.01799 -7.92 1.60e-13
## 5 imp2 (Intercept) 6.44744 0.14350 44.93 8.28e-106
## 6 imp2 log(GDPperCapita) -0.20265 0.02722 -7.45 2.91e-12
## 7 imp2 contraception -0.00596 0.00206 -2.90 4.18e-03
## 8 imp2 educationFemale -0.13358 0.01461 -9.14 7.46e-17
## 9 imp3 (Intercept) 6.57374 0.15260 43.08 3.79e-103
## 10 imp3 log(GDPperCapita) -0.20811 0.02774 -7.50 2.00e-12
## 11 imp3 contraception -0.00507 0.00224 -2.26 2.48e-02
## 12 imp3 educationFemale -0.14579 0.01707 -8.54 3.42e-15
## 13 imp4 (Intercept) 6.49875 0.17864 36.38 1.29e-89
## 14 imp4 log(GDPperCapita) -0.21912 0.03250 -6.74 1.68e-10
## 15 imp4 contraception -0.00710 0.00228 -3.11 2.14e-03
## 16 imp4 educationFemale -0.11895 0.01596 -7.45 2.75e-12
## 17 imp5 (Intercept) 6.52708 0.16142 40.44 1.24e-97
## 18 imp5 log(GDPperCapita) -0.21800 0.03058 -7.13 1.84e-11
## 19 imp5 contraception -0.00650 0.00218 -2.98 3.27e-03
## 20 imp5 educationFemale -0.12604 0.01722 -7.32 6.09e-12
Conducting inference
## term estimate std.error estimate.mi std.error.mi
## 1 (Intercept) 6.8840 0.29039 6.50402 0.16896
## 2 log(GDPperCapita) -0.2943 0.05765 -0.20999 0.03097
## 3 contraception -0.0113 0.00424 -0.00588 0.00247
## 4 educationFemale -0.0770 0.03378 -0.13338 0.02064
Missingness map
